"Boolean" operation on bézier circles
Use Boolean modifier after adding some thickness to circles so modifier to work or use Knife Project.

Cutting with Boolean modifier

Boolean didn't work in your case most likely because there wasn't geometry needed for it to operate with. Two flat faces won't be cut with any Boolean operation; if you extrude these faces in any direction (e.g. by Z axis) and then add the Boolean, it will be executed correctly. 


First fill the circles with geometry: if they are still curves convert them to 2D; if they are meshes select all the vertices and press F. Position them accordingly: center one should be as separate object, and 3 others should be joined as another one.
Extrude them / add Solidify modifier; you may leave small values in order the extrusion to be deleted further on as double vertices (increase threshold a bit to achieve that). 
Finally add Boolean to the center circle and choose object with 3 circles as a target. Delete created additional geometry if needed.




Knife Project

It's basically described here. In this case you shouldn't extrude anything. Some general steps:


Place circles as in the 1-st way; 
Select those 3 joined as the first object, then one in the center and enter their Edit mode (the selection order is important because it determines where faces will be cut from).
Press  Numpad 7 to enter Top View. It will make Knife Project to cut object using this projection.
In the Toolshelf (open with T) find Knife Project and choose it.




See also:

Cases where boolean operations can fail

How would you cut a shape out of an object using another object?